Bank Valuation New Approaches

A new approach to the valuation of banks Michael Adams∗ , Markus Rudolf∗ This version: November 24, 2006 Abstract Despite the discussion surrounding the disintermediation of the financial sector, banks still play a prominent and special role in the allocation of capital in the economic system. We argue that the business model of the bank exhibits such peculiarities that it deserves a special treatment in the approach to its valuation as well. In particular, the exposure to interest rate risk is a major characteristic of its business, not only in the process of maturity transformation but also as a major determinant of price margin and business volume. Although these traits have been noted before in the literature, to our knowledge, there exists no common framework to value a bank which adequately accounts for these features. We propose a valuation model for banks based on Merton’s (1974) structural model of the firm, which we adapt to the banking firm by the help of term structure models of the interest rates. In this setting, we interpret banks as a particular portfolio of long and short positions in interest-rate sensitive assets with special characteristics, and thus benefit from the rich toolbox of continuous-time finance to derive its value. In doing so, we are able to show another peculiarity of bank valuation, i.e. that the exercise price of a call option on the firm value, representing the bank’s equity, is not the face value of bank liabilities but their economic value. JEL classification: C22, G12, G13, G21. WHU, Dresdner Bank Chair of Finance, Burgplatz 2, 56179 Vallendar, Germany. Contact e-mail: michael.adams@whu.edu. We are grateful to Stewart C. Myers, Matthias Muck, and participants of the 2005 Burgenland doctoral seminar for helpful comments and the WHU USA foundation for its financial support. Any remaining errors are our own. ∗ 1 1 Introduction The principal function of financial markets is the efficient and intertemporal allocation of capital, but, if let alone, financial markets tend to imperfections and exhibit frictions in performing this function. Financial intermediaries in general, and banks in particular, owe their existence to these inefficiencies in the distribution of capital from those with a surplus to those with a need, i.e. reducing transaction costs, and at the same time efficiently selecting among those with capital needs according to their respective risks, thereby solving problems of asymmetric information. Most of the reasons literature has brought forward for the existence of banks can and have been subsumed under these two broad categories.1 The importance of these services has been underpinned in several studies showing that the level of development and sophistication of financial intermediaries, among which banks certainly belong to the most important, can have a significant impact on economic growth.2 When we define a bank’s business as accepting (shorter-term) deposits and issuing (longer-term) loans in our framework, this is congruent to common definitions in the literature.3 As such, the bank is one of the oldest institutions of financial intermediation and still plays a prominent role in the economy for the allocation of capital—this also in spite of the discussion surrounding the disintermediation of the financial sector.4 Nonetheless, technological advances have had a profound effect on banking and have lead to a worldwide consolidation in the sector.5 In Germany, consolidation is also underway but still lagging behind. Germany’s comparably still very fragmented banking market is often attributed to the institutionalized segregation in three pillars.6 However, the strict segregation of this three-pillared system is currently up for discussion, possibly opening the gates for further consolidation in the largest European banking market. Given these considerations, the value of banks is clearly a question of interest, be it for shareholder value-oriented management or in the course of a merger or acquisition. Although firm valuation is one of the core problems of corporate finance and has attracted extensive coverage in the literature, we argue that a bank’s See e.g. Santomero (1984) or Bhattacharya and Thakor (1993) for surveys and Freixas and Rochet (1997) as a more comprehensive textbook. 2 For a review of this issue, see e.g. Levine (1997). 3 See e.g. Freixas and Rochet (1997). In a classification of banks, this refers mainly to commercial banks, but not exclusively. For example, in the regulatory framework of many countries, a commercial bank is clearly distinct from e.g. a savings institutions or a savings and loan association. However, all three would fall within our definition of a bank as an institution issuing loans financed by deposits. The facts that e.g. thrifts in the U.S. mainly issue loans in the form of mortgages, or savings institutions in Germany are obliged to serve common welfare, are technical details with little relevance for our purposes. 4 See e.g. Schmidt et al. (1999) for a study countering the common arguments of disintermediation and supporting the importance of banks. 5 See e.g. Berger et al. (1999) and a May 2006 special issue of The Economist titled “Thinking big: A survey of international banking”. 6 See Hackethal (2004) for an overview and Decressin et al. (2003) for a detailed analysis of the German banking system. 1 2 business exhibits peculiarities that do deserve a special treatment in the approach to its valuation. Although this problem obviously does not require a separate and novel pricing theory, certain deviations from standard methods seem not only appropriate but also necessary to us; to say it with Damodaran’s (2002) words: “The basic principles of valuation apply just as much for financial service firms as they do for other firms. There are, however, a few aspects relating to financial service firms that can affect how they are valued.”7 We will demonstrate both why and how standard valuation approaches should be modified to account for bank-specific business risks. Specifically, we address interest rate risk as a major characteristic of the banking business. The bank faces this type of risk not only directly in the process of maturity transformation but also indirectly in the determination of price margins and the attraction of business volume, all affecting the value of the bank. The high interest-rate sensitivity of a bank’s market value has been noted many times before in the literature; for example, early empirical studies supporting such a relationship are Flannery and James (1984a) for U.S. banks and Bessler and Booth (1994) for German banks. Accordingly, there exist several authors who have taken up this finding and suggested special bank valuation approaches. However, in the process of reviewing these approaches, we will show that their models do not reach very far in incorporating bank-specific risks. Hence, to our knowledge, there exists no common framework to value a bank which adequately accounts for these features. We propose an alternative and new valuation model for banks based on term structure models of the interest rates. These models allow us to directly account for interest rate risk but let us avoid the problem of having to explicitly forecast interest rate developments, a problem inherent in all other approaches. Originally, term structure models were developed for the valuation of interest-rate sensitive derivatives. In the same spirit, we model banks as a portfolio of interest-rate contingent claims and value bank equity as a call option on the portfolio value. In other words, we build on Merton’s (1974) structural model of the firm and extend its standard asset process of the firm for an application to the dynamics of the banking firm. For expository reasons, we will set up our model first in discrete time to gain an intuition for the bank’s business model and will then derive a valuation model in a continuous-time setting. The paper proceeds as follows. In Section 2, we review the existent literature on bank valuation and identify shortcomings of the present approaches. In section 3, we review the bank’s business model and motivate a distinct approach to its valuation. We feel that this is necessary given the fact that this problem has attracted little attention in the literature so far. In the fourth section, we sketch a simple version of our model in discrete time and in section 5, we extend this model to the mathematics of continuous time in order to derive a solution. Section 6 concludes. 7 Damodaran (2002), p. 603. 3 2 2.1 Survey: Literature on bank valuation Why banks are special Little has been written on the valuation of banks. This fact allows for two conclusions: Either this question is of minor relevance and the valuation of banks deserves no special attention, or, alternatively, bank valuation exhibits particularities and problems that have not found appropriate attention in the literature so far. In seizing this problem, we follow the latter view and rely on the few existing papers contributing to this issue and on those explicitly mentioning the valuation of banks as one of the unresolved issues in financial research, as e.g. Copeland et al. (2005) do.8 In corporate finance, it is not unusual to specify valuation models for particular types of firms. For example, to mention just two of them, Brennan and Schwartz (1985) propose a real-options based valuation approach to natural resource companies, explicitly modeling the options to temporarily close, reopen and shut-down the mine, depending on the market price of the resource and Kronimus (2002) develops a model suitable to the traits of young, fast-growing firms, including little or negative current earnings but fast revenue growth. These models do not aim at introducing a new paradigm in asset pricing theory. Rather, the common feature of these models is the attempt to better grasp the underlying characteristics of the business—on which equity is the residual claim—as when compared to standard approaches. For the same reasons, one can argue for a special valuation approach for banks. The characteristics of the banking business motivating a distinct valuation approach can be subsumed in four categories. First, due to their central role for the economy, banking is typically a heavily regulated industry, covering a wide range of provisions, such as e.g. market entry, deposit insurance, reserve requirements, or capital structure.9 Second, banks operate on both sides of their balance sheets, actively seeking profits not only in lending but also in raising capital, a duality which had been practiced for long but had not been fully understood by economists until the late 19th century: “A Banker is a trader whose business consists in buying Money and Debts, by creating other debts.”10 From a financial accounting point of few, this implies relatively few fixed assets, resulting in a low operating leverage, and relatively high financial leverage, resulting in a comparably higher earnings volatility.11 Third, and as a consequence of the previous point, banks are exposed to credit default risk, but in contrast to other firms, they also actively seek this kind of risk as part of their business model. Last but not least, the profit and the value of the bank is strongly dependent on interest rate risk. Tackling each of these four points is of unequal complexity. Solving the first See Copeland et al. (2005), p. 872. See e.g. Carey and Stulz (2006). 10 MacLeod (1875), p. I:275. 11 It can be shown that banks optimally operate at high leverage because of deposit insurance schemes granted to them in almost all industrialized countries, which is essentially a put option given to the bank, see e.g. Buser et al. (1981). 9 8 4 issue is straightforward, since the regulatory framework is usually known and the same for all banks. Regulations have to be accounted for, but they merely impose deterministic structural restrictions on a valuation model. Similarly, the second peculiarity does not pose large problems either. Instead of applying the entity method, which measures the cash flows to all claimants of the firm and discounts them at the weighted-average cost of capital, one has to rely solely on the equity method, treating interest expense as a cost, more precisely an operating cost in the case of banks. Adequately measuring the credit default risk of a bank’s assets is a more challenging issue. Nonetheless, its effect on bank value is obvious—a downside risk proportional to the credit risk of the loan portfolio—and one can readily apply models of credit portfolio management to account for its effect on the value of the entire bank.12 Much more interesting are the effects of interest rate risk on bank value, since they surface at many different corners of the bank’s business. Most obviously, the slope of the term structure determines the profitability of the maturity mismatch between a bank’s assets and liabilities. In addition to this,there exist further and more subtle effects, though. For example, bank rates adjust sluggishly to changes in market rates, and do so asymmetrically, i.e. the speed of adjustment is different when rates are rising as when they are falling, resulting in a time-varying spread which is typically larger in a high interest-rate environment. These empirical traits have been documented in several studies, such as e.g. Hannan and Berger (1991), Ausubel (1991), or Neumark and Sharpe (1992), and current data in the latest monthly report of the Bundesbank (2006) indicates similar evidence. Besides, the demand for deposits and loans depends on both, the market rate and the rate a bank charges or pays, respectively, relative to the market rate. Also, loan demand and deposit demand typically have a negative correlation. Taken together, the effects of interest rate risk on bank value are significant but nontrivial. Samuelson (1945) is among the pioneers to argue for the (positive) dependence of a bank’s value on the level of interest rates. However, for several decades, his purely theoretical reasoning has not been adequately matched with supporting empirical results. Contemporary surveys on the interest-rate dependency of bank’s equity value typically start with Flannery and James (1984a, 1984b), whose empirical studies could confirm Samuelson’s earlier suggestions. For a sample of U.S. banks, Flannery and James (1984a) find that bank stock returns and interest rate changes are highly correlated. They introduce the interest rate as an additional risk factor in a two-factor intertemporal capital asset pricing model, where the investment opportunity set varies with the level of interest rates, as first suggested in Merton’s (1973a) ICAPM. Furthermore, their results are consistent with what they call the maturity mismatch hypothesis, i.e. that cross-sectional variations in interest rate sensitivity result, at least in part, from differences in the assets and This is not to mean that measuring credit portfolio risk is straightforward. Rather, we imply that there are already abundant suitable models for this task, a review of which would deserve a paper in its own right. See e.g. Crouhy et al. (2000) or Uhrig-Homburg (2002) for a survey of credit risk models. 12 5 liabilities structure. In addition, Flannery and James (1984b) show that the interest rate sensitivity does not depend on the stated maturities but on the effective maturities of instruments on the bank balance sheet. Usually, the bulk of bank liabilities consist of core deposits, to which we will refer in the following simply by “deposits”. This is for ease of exposition and is not to be confused with the general meaning of deposits, which also includes deposits with a pre-defined maturity, for example certificates of deposits (CDs) or time deposits.13 These (core) deposits have a very short or no maturity but their effective maturity tends to be longer, which can be concluded from two observations: First, although depositors are allowed to withdraw their balances every day, the “half-life” or “decay rate” of balances tends to be significantly longer; second, banks take advantage of this sticky behavior and adjust bank rates in response to market rate changes only sluggishly, especially in times of rising interest rates. Consequently, a higher share of core deposits in bank liabilities results in a decrease of both the maturity mismatch and interest rate sensitivity of bank stock returns, ceteris paribus. In the following, many authors refined these methods and added new aspects, of which the overwhelming part found supporting evidence consistent with Flannery and James.14 However, in concluding it should be remarked that these approaches draw a fairly general picture, since the inclusion of a second interest rate risk factor in a multi-factor model only accounts for the the interest-rate related business risk of the banking industry in general. 2.2 Existing approaches to bank valuation Interestingly, one finds mostly German authors who take up the problem of valuing banks, with many Ph.D. theses published on this topic, whereas in the AngloAmerican literature, we are only aware of textbooks and guides for practitioners, although the previously mentioned Copeland et al. (2005) list it as one of the unresolved problems of financial theory. Among the German authors, early treatments of Zessin (1982) and Adolf et al. (1989a, 1989b) were followed by several theses during the 1990s on the shareholder For a more detailed characterization, it can be said that in the U.S., four types of deposits are characterized as non-maturing core deposits: demand deposit accounts (DDAs), negotiable order of withdrawal (NOW) accounts, money market deposit accounts (MMDAs), and passbook accounts. DDAs are standard accounts held for transaction purposes and represent the greatest part of money supply in the U.S. NOW accounts are savings accounts that serve for transaction through negotiable orders of withdrawals, on behalf of the holder and payable to third parties. MMDAs pay a higher interest rate than traditional deposits but are limited in the number of transactions they allows per month. Passbook accounts have a legally required short notification period before withdrawals. 14 For the U.S. market, these include e.g. Saunders and Yourougou (1990) and Elyasiani and Mansur (1998). For Europe and Germany in particular, see for example , Bessler and Booth (1994), Oertmann et al. (2000), or Bessler and Opfer (2003). A particularly interesting study is that of Kane and Unal (1990), who identify off-balance sheet sources why bank stocks might seem insensitive to interest rate shocks while in fact they are not. This reconciles contradicting results of several previous studies such as Lynge and Zumwalt (1980), Chance and Lane (1980), or Kane and Unal (1988). 13 6 value of banks, including K¨mmel (1995), Behm (1994), H¨hmann (1998), and u o Kirsten (2000), and by one thesis of Sonntag (2001), which deals exclusively with the particularities of bank valuation. Textbooks and practitioners’ guides include e.g. Miller (1995), Johnson (1996), Copeland et al. (2000), Damodaran (2002), and Koch and MacDonald (2005). Irrespective of their origin, the reasons all these bring forward for a special bank valuation approach largely fall within the four categories mentioned in the preceding section, among which, in accordance with our remarks in the previous section, most stress the interest rate risk as one of the most important variables affecting the value of bank equity. Although Zessin (1982) emphasizes the interest rate risk as a major determinant of bank equity value, his propositions to account for it are rather crude. First, he assumes that the level of interest rates is mainly driven by macroeconomic factors and proposes expert surveys and macroeconomic outlooks as basis for an interest rate forecast. Then, assuming that there is little room for individual pricing behavior in the oligopolistic banking market, he suggests a regression analysis to determine the relationship between bank rates and market interest rates. Finally, this method should be repeated in a scenario-like analysis.15 A serious drawback of such an approach is the fact that the predictive power of expert surveys is often disappointing; of the many studies coming to this result, see e.g. Brooks and Gray (2004). Further, there is little room for cross-sectional and intertemporal differentiation among banks. Last but not least, Zessin mentions the interrelated problem of forecasting business volume on the asset and liability side on one hand and respective interest margins on the other, but offers little to overcome it.16 In the remainder, he discusses further aspects in which banks distinguish themselves from other firms, though the overall valuation procedure is of little difference to firm valuation methodologies. Adolf et al. (1989a, 1989b) also suggest a combination of forecasts of the interest rate and business volume, and scenario techniques to grasp the influence of interest rate risk on bank earnings, and gradually refine the previous approach in that they look at strategic business units instead of the bank in its entirety. They introduce a “maximum payout assumption”, according to which the bank retains cash flows only up to the regulatory capital required for its business in the next period and pays out all remaining cash flows. Additionally, they suggest the application of an instrument from internal bank controlling, the transfer pricing based on matchedmaturity marginal value of funds (MMMVF), in the valuation process.17 In so doing, they are able to separately value the asset business, the liability business and the treasury. Such a procedure offers several advantages, most notably the consideration of different risk factors for the asset and the liability business, and the possibility to isolate the interest rate risk from maturity mismatches in the treasury. Zessin (1982), p. 121–127. Later, he includes a description of several instruments to measure interest rate risk, but does so only in a separate analysis, i.e. not incorporating these methods in the valuation process. 16 See Zessin (1982), p. 126. 17 Their adaption of the MMMVF is the “Marktzinsmethode” as outlined e.g. in Schierenbeck (1997). We will return to this model in more detail in the following section. 15 7 This idea is further outlined and worked on in Behm (1994) and K¨mmel (1995), u of which the latter elaborates on alternative forecasting methods for business volumes, interest rates and, depending thereon, bank rates.18 He focuses clearly on the development of interest rates as a key driver of bank value, adding further complexity with his point that the bank rate is not a passive function of the market rate but also dependent on management decisions and business strategies.19 To account for the effects of interest rate changes, he proposes another tool of bank controlling, the balance sheet of interest rate elasticities.20 Without delving into a detailed description here, one can characterize this tool as offering the advantage to include the “responsiveness” of bank rates to changes in market rates, which is not accounted for in purely market-based measures of interest rate risk such as e.g. duration. Although this offers a potential enhancement in accuracy and elasticities seem to be relatively stable,21 it merely shifts the forecasting problem since it implicitly assumes that other market parameters, for example intensity of competition or customer behavior, remain constant over time as well. Ultimately, K¨mmel’s (1995) approach does u not offer alternatives to forecasting interest rates, and for this he relies on the same methodologies as previous authors, implying the same serious drawbacks. Sonntag (2001) also builds his model upon the transfer pricing of MMMVF; his innovation is the valuation of bank cash flows based on the method of certainty equivalents.22 Not surprisingly, the practitioners’ guides offer little methodological innovation. Their focus is more applied in nature and their suggestions seem to be easier to implement. According to them, the two most prominent problems in bank valuation are to determine the quality of loans and to understand the role of interest rate risk and maturity mismatches. As to the former, they discuss the problems for outsiders to appropriately value the loan portfolio, which is further severed by managerial freedom in the share set aside for loan loss provisions; for the latter, they also propose a model based on internal transfer prices. For example, Copeland et al. (2000) compare a net income model for non-financial firms with a spread model specifically for banks, where the latter is largely equivalent to the transfer pricing model based on MMMVF mentioned above. In so doing, they summarize the advantages of the spread model in explicitly requiring a forecast of four parameter: (1) the spreads between respective bank rates and market rates, and their elasticity to changes in market rates, (2) the dynamics of inflows and outflows of funds over time and at respective rates, (3) the substitution between bank products as interest rates change, and (4) the portion of profits purely due to maturity mismatch and how sustainable it is over time.23 Nonetheless, they admit at the same time that “[i]t is not easy to See K¨mmel (1995), p. 59 and pp. 66–98. u See K¨mmel (1995), p. 71 and p. 97. u 20 The “Zinselastizit¨tsbilanz” as proposed e.g. in Rolfes (1999), part 3. a 21 See e.g. Rolfes and Schwanitz (1992). 22 He further claims that one of the main results of his treatise is to have shown the irrelevance of banks in the business of maturity transformation if one remains in a neoclassic framework, a claim that we find hard to approve. See e.g. the rich literature on the existence of banks as quoted in the beginning. 23 See Copeland et al. (2000), p. 434. 19 18 8 build all of these variables into a forecast” and do not offer alternatives to interest rate forecasts, either. To summarize these works, it can be said first, that there exist reasons why bank valuation represents a problem that is distinct in certain respects from firm valuation; second, that these reasons can be grouped in four categories; third, that among these four problems, interest rate risk is the most important one; and fourth, that the methods proposed to account for this kind of risk are unsatisfactory and leave room for improvement. 3 Stylized business model of the bank A bank’s valuation begins with understanding its business model, which reflects its operating activities, sources of revenues and costs structure. Once having established a suitable business model in this section, we will value the bank’s equity as a call option on the bank’s asset value in the next, following the insights of Merton’s (1974) structural model of the firm. The Merton model provides a risk-neutral valuation framework within which precise solutions for the valuation of various financial claims on the firm (or asset) value can be derived. This precision comes at a price and indeed merely shifts the ambiguity of firm valuation to the specification of the business model that generates the firm value. Hence, in this section we characterize the bank’s business model and in so doing lay out the foundation for a model of the banking firm value. A bank is “an institution whose current operations consist of granting loans and receiving deposits from the public.”24 As such, it seeks to make a profit on both sides of the balance sheet, a fact that is sometimes referred to as duality of the banking business. For now, and without loss of generality, we abstract from taxes, reserve requirements and other sources of income; further, we will also neglect default risk of bank customers for the beginning. Then, a bank’s profit, π, can be stated in its most simple form as difference between the return on assets and the costs of liabilities, net of general and administrative costs, C, π = rA × A − rL × L − C = N II − C. (1) Here, the difference rA × A − rL × L is also known as net interest income, N II. Obviously, the factors determining the current N II are the realized spread and the existing business volume, which can expressed in terms of average interest-earning assets as we abstract from reserve requirements.25 Future N II is driven by the expected spread, the expected growth in interest-earning assets, and additionally the development of the term structure of interest rates, as bank assets and liabilities typically differ in maturity and composition of fixed and floating rate products. Freixas and Rochet (1997), p. 1. Accounting for reserve requirements would result in a deduction from liabilities and hence a difference between interest-earning assets and interest-bearing liabilities. Introducing such a deterministic feature later does not pose a problem but doing so now would further complicate our exposition at this point. 25 24 9 A theoretical model for the exercise of market power by the banking firm in an imperfectly competitive Cournot-type of market has first been established by Klein (1971) and Monti (1972), who formalize the idea of a positive relationship between the relative NII and the market power of a bank, ceteris paribus.26 The assumption that banks operate in an imperfectly competitive market is intuitive. In the liability business, banks offer significant economies of scale in storing valuables and in providing an efficient access to the payment system. In the asset business, banks offer economies of scale in credit risk assessment, loan procurement and creditor monitoring.27 Naturally, the larger their customers are, the lower will be the advantages banks can offer in terms of economies of scale and accordingly, the lower their market power will be. For example, while it would not make sense to access capital markets for a single consumer loan, corporate customers will often find it cheaper to issue bonds instead of financing their projects by bank loans. Another important measure is the ratio of the NII to total income. Following these thoughts, this ratio can also take on two interpretations. First, it is clearly a measure for the interest-rate dependency of a bank’s income—the more income a bank can generate from service fees and provisions, the less it will depend on interestrate sensitive income. Second, this ratio can also be seen as a proxy for the customer groups a bank services and the market power it can exercise with them. While the latest monthly report of Bundesbank (2006), gives an overview of this measure for aggregate groups of German banks,28 let us pick out just two of them, Hamburger Sparkasse (Haspa) and Deutsche Bank (DeuBa), to make our point more obvious. Haspa is the largest German savings institution in Germany and mainly serves retail customers and regional SMEs; in retail banking, it has a local market share above 70%.29 On the other hand, DeuBa is the largest German bank and operates as a fully-integrated universal bank, offering commercial banking, investment banking, and asset management on a global scale and under one roof.30 While DeuBa’s commercial banking unit also serves retail clients, its corporate clients will typically be larger than those of Haspa. Not surprisingly, Haspa’s net interest income accounted for 62.7% of its “gross profit” in 2005, whereas this figure is only 23.4% in the case of DeuBa.31 Certainly, the difference in this ratio reflects the underlying differences in banking services offered. However, it is also conceivable that DeuBa is just not able to extract an adequate NII from its corporate customers and therefore has to “Relative NII” is to mean the net interest margin of a bank, which one obtains by dividing the net interest income by the amount of average earning assets. For a comprehensive summary of the Monti-Klein model, see e.g. Freixas and Rochet (1997), section 3.2. 27 These are also the economic justifications brought forward for the existence of banks. See e.g. Allen and Santomero (1998) for an overview. 28 See Bundesbank (2006), p. 23. 29 See company information at www.haspa.de. Haspa is the largest savings institution in terms of assets. 30 See company information at www.db.com. DeuBa is the largest German bank both in terms of assets and market value. 31 Net interest income is presented in % of total income net of interest expense in the denominator. This denominator can be likened in some sense to the gross profit of a non-financial firm. Figures for calculation are taken from the annual reports for 2005. 26 10 Value of bank equity as an interest rate contingent claim A = f (At-1, [rA, τ=2 – r] , XA) L = f (Lt-1, [r – rL, τ=1] , XL) VBANK = NPV (rA, τ=2 A – rL, τ=1 L – C) rA, τ=2 = f (rA, τ=2, t-1, r, Xr ) A Fristentransformationsbeitrag rL, τ=1 = f (rL, τ=1 , t-1, r, Xr ) L Figure 1: The stylized valuation model of the bank Michael Adams Neue Ansätze zur Bewertung von Banken 1 turn to other services as well. How do these insights bring us any further? They do so in two respects. First, the market power is a central feature of our bank model. Therefore, it is important to grasp its importance for the bank value, which is also underscored by the sheer number of studies analyzing this question.32 Second, since our model is primarily concerned with the interest rate risk in the business of commercial banks, this segmentation of banks offers helpful background information on the model’s relevance. Further, in such a segmentation, it should be stressed that our model can be applied to the valuation of universal banks as well, this for valuing the commercial banking unit in a sum-of-the-parts approach.33 The basic idea of a N II-based approach is depicted in figure 1. There, the bank’s equity value VBAN K results as the net present value of the N II net of costs, as long as we neglect items like depreciation, amortization, and changes in working capital so that the net income can be used as a proxy for cash flows. Besides, the figure lists already some of the variables that affect the constituents of the N II; for example, the asset volume A is assumed to be a function of its past value At−1 , the spread the bank charges above comparable rates, rA − r, and further yet unidentified factors summarized in the vector X A . Although this framework is suitable and has been proposed in several works on bank valuation, it also has serious drawbacks. A major weakness of valuing a bank based on the overall N II as shown in figure 1 is that it ignores differences in the maturity composition of assets and liabilities and the fact that banks are typically exposed to a positive duration gap, i.e. that the average There exist numerous papers analyzing almost as many factors impacting the N II relative to asset volume, see e.g. Ho and Saunders (1981), McShane and Sharpe (1985), Kolari et al. (1988), Allen (1988), Zarruk (1989), Moore, Porter and Small (1990), Ausubel (1991), Hannan (1991), Hannan and Berger (1991), Neumark and Sharpe (1992), Calem and Mester (1995), Scholnick (1996), Heffernan (1997), Angbazo (1997), Wong (1997), Sharpe (1997), Berger and Hannan (1998), Demirg¨¸-Kunt and Huizinga (1999), Kahn et al. (1999), Saunders and Schumacher (2000), Berger uc et al. (2000), Lim (2001), English (2002), Stanhouse and Stock (2004), Hanweck and Ryu (2004), Maudos and de Guevara (2004), Brewer III and Jackson III (2006). 33 The sum-of-the-parts approach is a common methodology for valuing a multi-product firm. See e.g. Copeland et al. (2000), ch. 14. 32 11 duration of assets is longer than that of liabilities. Yet this composition accounts in large parts for the changes in the N II in reaction to changes in the term structure of interest rates. Also, a bank’s market power in issuing loans might be much different from that in issuing deposits, a difference which remains undetected in this setting. Considering these problems, we follow a common practice in bank management and attribute the profits of entire bank to its constituent business units based on MMMVF-transfer prices, i.e. the opportunity cost—or, more appropriately, the opportunity rate—of an investment in bonds with a comparable risk profile.34 Although these transfer prices were originally developed as a tool of bank controlling, they can serve useful purposes in bank valuation and have been applied to bank valuation before, as mentioned in the previous section reviewing the literature. Furthermore and again simplifying, we assume that a bank consists of three business units, an asset business unit (ABU), a liability business unit (LBU), and a treasury department which is in charge of the asset-liability management of the bank (ALM). The ABU is in charge of issuing loans and, principally, of measuring and managing credit risk, although we abstract from any issues of credit risk in our model so far. The LBU issues transactions and savings deposits, providing its customers with an access to the payment system and the bank with an internal funding of the ABU. The ALM mainly has two task in our simple model of the bank. First, it has to equalize any imbalances between assets and liabilities with external financing, taking long or short positions in the interbank market.35 Second, it is in charge of hedging interest rate risk, most notably term structure risk stemming from the duration gap between the two sides of the balance sheet. Given this internal bank structure and further assuming that there exists market rates of various maturities τ at which the bank can lend and borrow in the interbank market, rτ , the profit in equation 1 can then be restated as π = (rA − rτ =2 ) × A + (rτ =1 − rL ) × L + (rτ =2 × A − rτ =1 × L) − C. (2) Here, rτ =2 signifies the market rate for a longer maturity and rτ =1 represents a comparable rate for a shorter maturity. Implicitly, this means of course that the respective bank rates have similar maturities. The market rates then serve as internal transfer prices of funds within the bank. In this way, the profit of the bank can be interpreted as the sum of the profits of the three business units defined above, net of cost.36 This conceptual view is depicted in figure 2. Thus, the profits of both ABU and LBU in this single-period model are conceptually driven by the same two factors as the N II of the entire bank: The spread See e.g. Rolfes (1999) or Dermine (2005). Although we allow the volume of assets and liabilities to differ within a period, with the difference offset in the interbank market, we rule out the case of permanent or even increasing deviations, i.e. that the two sides of the bank’s balance sheet have different long-term growth rates. 36 It is desirable to spread costs on the level of business units already. While this represents a sound theoretical approach, we are aware of the fact that there might be more than one viable way to split up costs between the units in an empirical implementation of the model. In the derivation of our bank value, we will largely ignore issues with costs. 35 34 12 BANK Asset Business Unit (ABU) rA, τ=2 Assets Asset-Liability Management (ALM) rτ=2 Assets and liabilities Liability Business Unit (LBU) rτ=1 Liabilities rL, τ=1 πABU = (rA, τ=2 – rτ=2) A πALM = rτ=2 A – rτ=1 L πLBU = (rτ=1 – r L, τ=1)L πBANK = rA, τ=2 A – rL, τ=1 L – C Figure 2: The stylized business units of a bank they can earn from their funds compared to the respective MMMVFs and the asset or liability volume of their business. In a multi-period model, we will need a third variable accounting for growth Fristentransformationsbeitrag Then, the bank derives its value in business volume. from three sources: Value of expected market power in the asset business and the liability business, and expected growth of the balance sheet. 4 Michael Adams Neue Ansätze zur Bewertung Risk-neutral valuation of von Banken bank the 1 Existing bank valuation approaches value these units based on their discounted cash flows and all run into the problem of having to forecast the development of the interest rate if they wish to consider its effects on cash flows. Relying on the same MMMVF-framework, we propose a different approach based on a structural model of the firm. Whereas Black and Scholes (1973) and Merton (1973b) remarked already that their option pricing theory can be applied to a variety of financial contracts, notably to the most basic claims in the form of debt and equity, it remained to Merton (1974) to develop a structural model of the firm in which he valued corporate debt as a contingent claim on the firm’s asset value, V , which he assumed to follow the standard Brownian motion dV = µdt + σdW, V (3) where µ is a drift factor and dW is a standard Wiener process. According to this approach, corporate debt can be valued as a portfolio of riskless debt and a short position in a put option on the firm’s asset, representing the default risk of the firm. The strike price of the call option representing firm equity and of the put option implied in firm debt are both the face value of outstanding debt. Since the put-callparity holds in this case, too, a structural model of the firm always implies a unified approach to the valuation of a firm’s equity and debt. Derivative pricing rests on the well-known principles of no-arbitrage and riskneutral pricing, according to which a hedging or replicating portfolio ought to have the same price tag attached as the prices of its constituent parts. The internal 13 hedging activities of the ALM and the hedging costs it charges to the other two units can be used to value the portfolios required to hedge the activities of the ABU and LBU. Then, these two units can be valued along the lines of the risk-neutral valuation approach. The activities of the remaining ALM are the simplest to value, akin to an long-for-short yield curve swap. Thus, we suggest to value the entire bank as a portfolio of interest-rate contingent claims with the value of bank equity resulting as a call option on such a portfolio. In this fashion, and over and above the advantages which we have already mentioned, our approach circumvents another common and nontrivial problem, i.e. the estimation of cost of equity for the three business units on a stand-alone basis. For this, and as conceptual link between our and a DCF-approach, it will be useful to measure the economic value of assets and liabilities, as opposed to their book values. In the proposed framework, the existence of a positive spread has the effect that the economic value of assets and liabilities does not equal its respective book value anymore. The relationship between the economic value of assets, VA , and liabilities, VL , to their respective book or face value, A and L, can be expressed as: VL = L − N P V (πLBU ), VA = A + N P V (πABU ), (4a) (4b) i.e. the economic value of a bank’s liabilities (assets) is its face value reduced (increased) by the net present value of the additional profits it is able to generate. The impact of equations 4 is intuitive: If a bank is able to charge a larger spread on its assets, their value will increase; similarly, if a bank is able to issue liabilities at a lower spread, this will also increase the bank’s value. One can clarify this even more talking the viewpoint of the hedging model based on MMMVFs. Then, since the excess-spread on assets is inversely related to the bond’s price, the ABU’s business can be likened to buying a bond below par from the bank’s creditors and selling it at par to the ALM, and inversely, the LBU’s business can be seen as buying bonds from ALM at par and selling them above par to depositors. The difference of both plus the result from maturity transformation is the residual claim of the bank’s shareholders, VBank , which they will only exercise if it is positive, of course. This implies the boundary condition VBank = max[N P V (πABU ) + N P V (πALM ) + N P V (πLBU ); 0] = max[VA + VALM − VL ; 0]. (5a) (5b) This is an interesting result and deserves further comment. Obviously, the exercise price of the option is VL , i.e. the economic value of liabilities and not their face value, L, as in the case of Merton’s (1974) model for non-financial firms. This highlights the difference between banks and non-banks and offers an intuitive interpretation: If the assets of a non-financial firm will not suffice to cover the firm’s notional liabilities, L, the shareholders will choose not to exercise their option and rather will default. In contrast, if a bank’s assets will not cover the face value of 14 its liabilities, L, shareholders might still be willing to exercise the call option in the boundary case. They will avoid a liquidation of the bank as long as the value derived from the liability business is large enough to cover the gap to the face value of liabilities: VBank = max[VA + VALM + N P V (πLBU ) − L; 0]. (6) Having established this basic boundary condition for the banking firm, we need to establish a stochastic process describing the evolution of the firm value as the underlying. For this, we have to identify variables affecting the value of the above portfolios representing the business units and the specification of processes describing the stochastic behavior of these variables. In our approach, we assume that our model bank offers just two products: A non-maturing deposit liability in its LBU and a a non-maturing overdraft facility in its asset business. Then, valuing a bank’s business units requires explicit modeling of three factors for each unit: 1. ABU and LBU: (a) Volume of assets and liabilities and their growth rates (b) Average effective maturities and durations of assets and liabilities (c) Changes in bank rates, both for bank assets and liabilities, in response to changes in comparable market rates 2. ALM: (a) Volume of assets and liabilities and their growth rates (b) Effective duration gap resulting from the difference between average effective durations of assets and liabilities (c) Dynamics of the term structure of market rates As to a bank’s balance sheet size and growth rate, it is possible to deduce estimates from past data; however, regarding the growth rate, this requires as well explicit assumptions on overall market growth rates and growth rates of competitors. The average effective maturities and durations also are at the core of bank value and, accordingly, its estimation is a key aspect of our valuation approach. While hedging against (parallel) shifts in the yield curve requires knowledge of the duration, this measure is based on an estimation of the effective maturity.37 Banks further distinguish themselves from each other in the spreads they charge on their assets above respective market rates and they pay on their liabilities below comparable rates. We call this the “market power” of banks. The sources of a bank’s market power can be manifold, one can think of it as e.g. the monopoly power of a regional bank in a geographically fragmented market or quality of service resulting in above-average customer loyalty. Although we will incorporate this feature in 37 See e.g. Mays (1997). 15 calibrating our model on past data and hence accept the market power of a bank as given, we should bear in mind that in applying our model to the long run, we have to additionally assume strategic factors that could erode or increase a bank’s market power. An additional relevant variable for the valuation of the ALM is the duration gap resulting as the difference of effective durations of the ABU and LBU. Since the ALM absorbs all the interest rate risk stemming from maturity transformation, it is also here that the dynamics of the term structure are relevant. The term structure model serves as a stochastic measure for interest rate risk and as such supersedes the problem of inaccurate forecasts of future market rates. Indirectly, the dynamics of the term structure will impact the value of all three business units, since we will model the asset and the liability rate of the bank as a function of market rates. The fact that we deal with two different maturities at the same time recommends the use of a two-factor interest rate model, which is better suited to grasp the dynamics of the yield curve. In such a two-factor model, the first factor is the short rate, just as in single-factor models, and the second factor is often chosen to be either a stochastic mean reversion speed or stochastic volatility. In comparison, single-factor models with constant volatility and reversion speed often exhibit poor fits on empirical data in many yield curve environments.38 One suitable two-factor model is the one proposed by Hull and White (1994), which offers computational advantages when compared to other models and allows an efficient implementation in discrete time, as Muck and Rudolf (2005) could show. Of course, such dynamics are the same for all market participants and as such do not represent a variable in which one bank can distinguish itself from others. Finally, it is obvious that these variables have many interrelations which we have to account for as well. For example, the effective maturity of a product will determine the appropriate market rate for transfer pricing, and changes in bank rates will have an influence on business volume. In the following, we begin sketching the valuation of the LBU, will then show how the valuation of the ABU mirrors the approach of the LBU and finally will turn to the ALM. 4.1 Valuation of the liability business unit (LBU) We mentioned already that typically, the largest portion of bank liabilities consist of core deposits, i.e. deposits that have either no or a very short maturity, thus representing the centerpiece of a bank’s liability side profits. Although deposits factually have no maturity, i.e. mature daily, and should therefore be very interest rate sensitive, they often are not and behave more like longer-maturing liabilities. This is attributed to the fact that the deposit rate is administered rate, i.e. set by the bank in an imperfectly competitive market. If one assumed that the stated maturity defined the effective maturity of deposits, this meant that deposits would See Rebonato (1998) for a discussion of theoretical and practical issues of various single-factor and two-factor models. 38 16 be withdrawn as soon as they matured, leaving little room for the premium earned by the bank. Empirical data contradicts this, as banks typically enjoy high retention rates for their deposits. For example, estimates for U.S. thrift institutions in studies performed by the Office of Thrift Supervision (1994, 2001) indicate that yearly retention rates for certain deposit types are above 75%. Hence, the assumption of a longer effective maturity is in accordance with observed behavior of depositors, lower interest rate sensitivity of deposit rates and volumes, and subsequently higher premium earned by banks on their liabilities. In this manner, the estimate for effective deposit maturities affects the value of the LBU in two ways. First, it determines the time horizon during which the bank is able to extract a premium on a given deposit base without attracting new deposits. Besides deposit volume growth, this is an important measure for the business volume of the LBU. Second, in our transfer pricing model of the bank, the appropriate market rate rτ =1 should be one with a duration comparable to the duration of deposits. Although duration and maturity are obviously different measures and “most banks confuse the length of time a dollar stays in the demand deposit (the maturity) with the sensitivity of balance values to changes in interest rates (the duration),”39 the close relation between the two is obvious as well: An increase in the deposits’ effective maturity due to a change in market power entails an increase in the effective duration of deposits. In turn, a larger duration of a bank’s deposits raises the duration of the transfer price rτ =1 at the same amount, which increases the profit of the LBU as long as the term structure is upward sloping. In financial research, the analysis of deposits as interest-rate sensitive claims is rather young. Following the development of the option pricing theory, a plethora of quantitative models for the valuation of interest-rate sensitive claims sprang up, whereas the valuation of deposits seemed to be the sleeping beauty and it took more than 20 years to kiss her awake. During the 1990s, several authors started to develop original approaches to their valuation, accounting for both interest rate risk and the problem of indetermined maturity. To our knowledge, these include Office of Thrift Supervision (1994, 2001), Selvaggio (1996), Hutchison and Pennacchi (1996), Jarrow and van Deventer (1998), and O’Brien (2000).40 Shortly reviewing these will certainly help us for our own approach and we will do so along the four key issues of deposit valuation: (i) The valuation approach chosen, (ii) the way in which interest rate risk is accounted for, (iii) the method chosen to model deposit volume and its growth, and (iv) the method chosen to model deposit rates. An overview of this comparison is shown in figure 3. Copeland et al. (2000), p. 439. Extensions to their approaches, most notably to the one of Jarrow and van Deventer (1998), were developed by Goosse et al. (1999), de Jong and Wielhouwer (2003), Sheehan (2004) (nicht in bib), Laurent (2004), and Kalkbrener and Willing (2004). 40 39 17 Interest rate Deposit volume risk (function of) 1 (1) OTS (1994, DCF /OAS-IRR Deterministic Past balance, market 2001) rate, deposit rate, retention rate (2) HP (1996) Contingent claims Vasicek (1977) Market rate, deposit (general equilibrium) rate, exogenous demand, trend gowth (3) Selvaggio DCF / OAS-IRR CIR (1985) Past balance, market (1996) rate, nominal income, seasonal dummy (4) JvD (1998) Contingent claims HJM (1992) Past balance, retention (no-arbitrage) rate, growth rate, market rate (5) O'Brien Contingent claims CIR (1985) Past balance, market (2000) (no-arbitrage) rate, deposit rate, (regional) income Model Valuation approach Deposit balance growth Deposit rate (function of) (function of) None Past deposit rates, interest rate, asymm. parameter Historical trend growth Market rate, deposit rate demand elasticity Market rate, past deposit rate, volatility parameter market rate, meanGrowth rate net of retention rate, market reversion to average deposit rates rate Historical trend growth market rate, meanrate reversion to cond. expected deposit Nominal income Figure 3: Overview of deposit valuation models Remarks: : Expected spot rates are inferred from the current forward rate curve; OAS-IRR: option-adjusted spread internal rate of return; CIR: Cox, Ingersoll, and Ross; HJM: Heath, Jarrow, and Morton. 1 4.1.1 Valuation approach The Office of Thrift Supervision (OTS) (1994, 2001) developed its “Net Portfolio Value Model” (NPVM) in 1994 and added an update to some of its equations in 2001.41 The OTS framework applies a discounted cash flows approach to the valuation of deposits. It estimates the cash flows generated by several deposit types on a monthly basis, net of costs. Instead of determining an appropriate discount rate for obtaining the net present value of deposits, the OTS uses price data from deposit sales or purchases as an input parameter. With this given as an estimate of the net present value, it discounts the cash flows with LIBOR plus a spread that equalizes the discounted cash flows to the value estimate. Hence, the NPVM can be likened to an internal rate of return analysis, where the value of deposits is expressed as an option-adjusted spread (OAS) above LIBOR.42 Selvaggio (1996) applies a valuation methodology similar to that of the NPVM, in that he estimates the OAS above the zero coupon rate that equalizes the discounted future cash flows of deposits to their empirically observed values. Hutchison and Pennacchi (1996) (HP) develop a contingent claims valuation model for deposits and are the only ones building their model on a particular equilibrium model of the economy which allows for the existence of banks. Specifically, Compared to the original approach, the basic structure and approach remained the same in the version of 2001, though, and the NPVM of both years will be presented as one model here. As the OTS is a regulatory and supervisory agency for thrift institutions in the U.S., the features and assumptions of this model, unlike those of the remaining models, have direct implications for the operation of U.S. thrifts. 42 LIBOR was introduced in the 2001 framework. The NPVM of 1994 used rates paid on secondary certificates of deposits. 41 18 they found their approach on Hutchison’s (1995) Cournot-type equilibrium model with imperfect competition. In this way, the spread of the deposit rate below the market rate is internalized as the optimal spread, i.e. the profit-maximizing spread based on the trade-off between price and volume as captured by the price elasticity of deposit demand—or deposit supply, to be precise. The value of the deposit business is then obtained as the discounted value of future oligopoly rents—or, more appropriately, oligopsony rents—where the rents are modeled as interest-rate sensitive claims. HP are able to derive a closed-form solution for their deposit valuation model. Jarrow and van Deventer (1998) (JvD) develop a contingent claims valuation framework for interest rate sensitive claims with exercise of market power in a noarbitrage setting and apply it to the valuation of demand deposits.43 They show that the valuation of these instruments can basically be understood as being akin to the valuation of a particular swap of which the principal is a function of past market rates. Applying the term structure model of Heath et al. (1992), they show how to obtain a basic closed-form solution for the pricing of deposits. Their work concentrates on the theoretical derivation of the model; an empirical implementation followed with Janosi et al. (1999) for the deposit data of one (anonymous) bank. O’Brien (2000) values deposits based on an arbitrage-free interest rate contingent claims framework which is similarly to the one of JvD and applies it to a data sample of NOW accounts and MMDAs of 74 U.S. commercial banks over 8 years.44 Like the NPVM of the OTS and unlike the models of JvD and HP, however, O’Brien allows for an asymmetric adjustment of deposit rates in accordance with empirically observed behavior. Specifically, in an environment of rising rates, the spread between market rates and deposit rates tends to increase whereas in times of falling rates, it tends to decrease. In the empirical implementation of his model, he finds the asymmetric feature in deposit rate adjustments to be statistically highly significant. However, due to the asymmetry he introduced, O’Brien is not able to derive a closed form solution and so has to rely on a Monte Carlo simulation of 1000 different paths. 4.1.2 Interest rate risk There is no interest rate risk inherent the NPVM. Given the forward rates as observed in the market, the OTS derives one static scenario of future spot rates on which all calculations are based. Selvaggio (1996) uses Monte Carlo simulation techniques and generates 300 interest rate paths to account for interest rate risk. He bases his simulation on the term structure model of Cox et al. (1985). While some parameters are taken from a previous estimation of this model by Chan et al. (1992), he calibrates the model on the LIBOR term structure of Eurodollar. HP model the market rate using an Ornstein-Uhlenbeck process as specified by Vasicek (1977). Although they derive their model for deposits, they state already that it can be applied in the inverse case to credit card loans as well. 44 NOW accounts are “negotiable order of withdrawal” accounts and MMDA stands for “money market deposit accounts”. See also a precedent footnote. 43 19 Since they are able to derive a closed-form solution to the valuation problem, no simulations are needed. Consistent with JvD’s arbitrage-free valuation approach, they model the term structure of interest rates according to the no-arbitrage approach of Heath et al. (1992). However, they show that, in principle, any other term structure model could be used as long as it specifies a SDE of the spot rate. O’Brien applies the underlying SDE of the market rate from Cox et al. (1985), i.e. an one-factor mean-reverting interest rate process which rules out negative interest rates. 4.1.3 Deposit volume and growth A common way to model deposits with no or short maturity is to assume that they are withdrawn as soon as they mature and that the bank then issues new deposits. The empirically observed stickiness of core deposit is accounted for by an autoregressive (AR) process of future deposit volume. For example, in a first-order autoregressive process, or AR(1) process, the deposit volume next period is largely determined by the volume in this period, Lt , Lt+1 = α + βLt + γ X + εt , (7) where γ is a parameter vector and X is a vector with other variables determining deposit volume, such as e.g. the level of interest rates, the spread between market rates and deposit rates, or nominal income. Most deposit valuation approaches apply such an autoregressive model and distinguish themselves in the relevant factors they identify and include, where the selection of relevant factors can have a significant influence on the dynamics of the calibrated process. A cautionary remark should be repeated: While such AR-processes are appropriate short-term forecasting, longerterm forecasts are more uncertain since these models take a market structure as given which might be due to change.45 In combining the elements of both stated and effective maturity, one can appropriately quantify the cash flow streams of deposit in a convenient fashion. Each period, t, the bank has cash outflows of consisting of previous period’s deposit volume plus interest claims thereon and cash inflows of consisting of new deposits. This pattern repeats itself up to the forecasting or valuation horizon T is reached and is exemplified in table 1. The length of t can be chosen in accordance with stated deposit maturity and data availability. t=0 +L0 t=1 +L1 −L0 −L0 × rL,0 +L0 × rτ =1,0) t=2 +L2 −L1 −L1 × rL,1 +L1 × rτ =1,1) ... ... ... ... ... t=T −1 +L(T −1) −L(T −2) −L(T −2) × rL,(T −2) +L(T −2) × rτ =1,(T −2) t=T −L(T −1) −LT −1 × rL,(T −1) +L(T −1) × rτ =1,(T −1) Table 1: Cash flow streams of the LBU 45 The same remark applies to the models of the deposit rate. 20 Similarly to deposit rates, the NPVM defines the level of deposit balances as a function of the spread between current deposit rates and market rates, the sensitivity of deposit rate to changes in the market rate, the institutions retention rate, and an industry-average of the retention rate. Additionally, the NPVM assumes that the difference between the last two factors converges to zero over time. Given its data on retention rates, the OTS assumes a maturity of 30 years for deposits. There is no growth assumption in the OTS framework, i.e. only the presently held deposit balances are valued, with levels declining as indicated by the retention rate, or respectively, its counterpart, the decay rate. Selvaggio specifies a partial adjustment model of deposits balance levels with the market rate, the nominal income, and a seasonal dummy as determining factors, yielding an in-sample fit of R2 = 98% for the four-year period between February 1991 and February 1995. Just as the OTS, he assumes a maturity of 30 years for his sample of demand deposits. Insofar as the functional form of deposit balance levels in this model depends on the nominal income, among other factors, there is an implicit growth assumption made, although it might be criticized for being rather vague. HP specify the equilibrium deposit balance level as a function of the market rate, the (optimal) deposit rate, an exogenous demand variable, and a trend growth rate. HP include this trend growth variable in the calibration of their model on empirical deposit data in order to captures the long-run equilibrium growth rate of the aggregate deposit volume. JvD model deposit balances as a function of the market rate, the change in the market rate from the preceding period, the previous deposit balance, and a time trend which is supposed to take up other (macroeconomic) variables not explicitly accounted for. Janosi et al. (1999) implement JvD’s model in specifying deposit balances additionally as a function of average past balance levels, a retention rate of old balances, a growth rate of new balances, and its sensitivity to changes in the market rate. Growth in deposit balances is accounted for in the model as described in the functional specification of deposit balances already. A retention rate measures the decay in old deposit balances and an exponential growth rate measures the increase in balances through new deposits. The growth rate is modeled to be sensitive to changes in the market rate. O’Brien (2000) specifies the level of deposit balances as depending on the current deposit rate, the current market rate, recent deposit balance levels, and a measure of nominal income which is interpolated of national data and regional data for the county where the bank is situated. Concerning the maturity of deposits, the author groups maturities and measures the contribution to the overall deposit premium along these maturity ranges. He finds that over 40% of premiums are contributed by maturities beyond 10 years, thus supporting the assumption of very long effective maturities of core deposits. O’Brien includes expected growth in deposits through a trend growth parameter. 21 4.1.4 Deposit rate behavior In the NPVM, deposit rates are a function of three factors, i.e. past deposit rates, the market rate, and a third factor accounting for whether the current deposit rate is above or below its long-term average on such deposits. With the last factor, the NPVM accounts for the asymmetry empirically observed in the adjustment of deposit rates. In this framework, deposit rates are a function of the market rate and in that way also deterministic. In Selvaggio’s (1996) methodology, the deposit rate results as the sum of the market rate and the OAS, which he calls the “demand deposit premium”. It is a function of past demand deposit premiums, the market rate as simulated in his Monte Carlo model, and an additional parameter that adjusts the OAS dependent on the level of LIBOR volatility. HP are able to derive optimal deposit rates as an endogenous function of the interest rate and the elasticity of demand for deposits. Both variables are simulated as mean-reverting Ornstein-Uhlenbeck processes and calibrated on historical data. As for deposit balances already, JvD specify deposit rates as function of the market rate, the change in the market rate from the preceding period, and the previous deposit rate. In the implementation of their model by Janosi et al. (1999), the deposit rate is calibrated on the data as a similar function, while adding additionally a mean-reverting function of its long-run average, thus adding the long-run average and the speed of adjustment as additional variables. In O’Brien’s (2000) framework, deposit rates are a mean-reverting function of past deposit rates and a conditional deposit equilibrium rate46 , an asymmetric adjustment feature as mentioned above, and market rates as obtained by the SDE. 4.1.5 Valuing the LBU Deposit valuation models are a sensible starting point to model the LBU’s value as part of the banking firm’s value. However, for our purposes only some of these approaches offer useful insights and modeling tools, and some adjustments have to be made. For example, our initial requirement of accounting for interest rate risk through term structure models is satisfied by four of the five approaches, namely Hutchison and Pennacchi (1996), Jarrow and van Deventer (1998), O’Brien (2000), and Selvaggio (1996), who all value deposits as interest rate contingent claims and account thereby directly for the effects of interest rate changes on deposit values. In contrast, the model of the Office of Thrift Supervision (1994, 2001) does not account for interest rate risk at all. Hutchison and Pennacchi (1996) nest their model within a general equilibrium framework of deposit banking, which allows additional economic interpretation but further complicates the model setting; likewise, it would take an extended equilibrium model to account for an additional equilibrium in the market of bank loans. As the author shows, when including asymmetric adjustment of deposit rates, the unconditional expected deposit rate and long-run expected rate deviate from the unconditional mean equilibrium rate. 46 22 Since this required extension to the general equilibrium model raises an additional issue for a complete and integrated framework of bank valuation, HP’s approach appears to be the more cumbersome than direct way to reach our ends. Selvaggio (1996), Jarrow and van Deventer (1998) and O’Brien (2000) seem to be the most promising approaches for our purposes and we follow their general valuation approach. In contrast to them, as we will see later, valuing bank assets and liabilities at the same time requires term structure models which allow for richer dynamics of the yield curve. Based on the previous models, the deposit valuation formula can be derived as a hedging portfolio of the cash flow pattern in table 1 in a risk-neutral valuation procedure, which yields47 T −1 VL,0 = ∗ P∗ E0 t=0 Lt (rτ =1,t − rL,t ) , g(t + 1) (8) P where E0 is the martingale expected value given the risk-neutral probability measure P∗ and g(t) represents the money market account with g(0) = 1. The deposit liability value based on the result in equation 8 can be likened to the value of an exotic interest rate swap, lasting for T periods, receiving floating at rt and paying floating at rL , and with an alternating principal of Lt . An alternative representation of this result is T −2 VL = L0 + P∗ E0 t=0 L(t+1) − Lt g(t + 1) T −1 − t=0 Lt × rL,t g(t + 1) − L(T −1) . g(T ) (9) for which JvD off an intuitive interpretation. According this restatement of equation 8, the value of the deposit liability is the sum of initial liability volume, plus the present value of any changes in volume over time, minus the present value of total costs, and minus the present value of deposit volume at maturity or valuation horizon. Hence, the deposit value is that of a series of T − 1 single-period, risk-free bonds paying below risk-free interest rates. Consequently, all of these bonds will have a price below par and shorting them can derive positive value, since the proceeds can be invested in the risk-free asset. This arbitrage transaction represents the liability business of banks. In the risk-neutral valuation framework, the LBU’s value is equivalent to the absolute value of costs of the hedging strategy that offsets the LBU’s expected cash flow streams. For the derivation of a valuation formula of the LBU, the functional representations of the deposit rate and the deposit volume can be substituted into either equation 8 or 9. 4.2 Valuation of the asset business unit (ABU) In our model, the only credit product a bank offers is a non-maturing overdraft facility. We have chosen this representative product for several reasons, one of the 47 See e.g. O’Brien et al. (1994). 23 being the similarity of its modeling to the modeling of deposits, and another one being that the rates a bank charges on its customers overdrafts are well in excess of the assumed credit risk, hence supporting our earlier argument of a bank’s market power. We chose this product for ease of modeling and exposition although, in principle, our approach can be applied to all kinds of credit products when allowing for their respective special characteristics and features. While there exists no liquid market for bank overdrafts, in the U.S. there is a market in asset-backed securities based on credit card loans, which in many respects are very similar to overdraft facilities. For example, Ausubel (1991) shows in a study on credit card loans that not only credit card interest rates are high and sticky but also that credit card loans change hands in interbank transactions at considerable premiums, often surpassing 20%. This indicates that banks earn disproportionately high interest rates on credit card loans on a risk-adjusted basis, i.e. after accounting for default risk. Indeed, in their deposit valuation model, Jarrow and van Deventer (1998) mention already the applicability of their approach to credit card loans, which we argue that this observation can in principle be extended to all kinds of bank loans. When neglecting credit default risk, obtaining a valuation formula for the loan business is a straightforward matter; table 2 and equation 10 are mirror images of the ones for the LBU. t=0 −A0 t=1 −A1 +A0 +A0 × rA,0 −A0 × rτ =2,0 t=2 −A2 +A1 +A1 × rA,1 −A1 × rτ =2,1 ... ... ... ... ... t=T −1 −A(T −1) +A(T −2) +A(T −2) × rA,(T −2) −A(T −2) × rτ =2,(T −2) t=T +A(T −1) +LA−1 × rA,(T −1) −A(T −1) × rτ =2,(T −1) Table 2: Cash flow streams of the ABU For simplicity of modeling, we have chosen the same time intervals as for the liabilities. Nonetheless, this a sensible assumption and only affects the frequency of steps in the autoregressive process, whereas the resulting effective maturity of assets should remain unaffected. Thus, the relevant rate would still typically be one of longer effective maturity, τ = 2. Assuming a loan business with one type of loan maturing t and a exogenously given loan volume of the bank, At , (compare also table 2) the economic value of the banks assets is analogously given by A(T −1) . g(T ) t=0 t=0 (10) We argued above that we abstract from credit risk in our valuation model. This assumption need not be detrimental to obtaining a realistic bank value. For example, we may assume that the rate a bank charges on a loan can, at least theoretically, be divided into two constituent parts, one representing the market price of credit risk and the other attributable to the bank’s market power in the loan market. An alternative method would be to assume that a constant fraction of the assets is VA = −A0 + P∗ E0 T −2 At − A(t+1) g(t + 1) T −1 + At × rA,t − CABU,t g(t + 1) + 24 lost, which according to Ausubel (1991) seems to be in accordance with empirical observations. More elaborate ways to include credit default risk of bank customers would be desirable but would also require to introduce more state variable. For not complicating matters further, we leave this aside for now. Without introducing more state variables at this time, we could include a conditional expected loss, EL(t), driven by our state variable, the interest rate. The return on the assets would then just be net of this expected loss. When adding this to 10, we obtain gives T −2 VA = P∗ −A0 + E0 t=0 At − A(t+1)−EL(t) g(t + 1) T −1 + t=0 At × rA,t − CABU,t g(t + 1) + A(T −1) . g(T ) (11) 4.3 Valuation of the asset and liability management unit (ALM) T −1 The ALM’s “residual” profit is represented by πALM = t=0 (rτ =2 × A − rτ =1 × L), (12) This reflects the idea that the ALM’s primary goal is not profit generation but managing the residual imbalance between asset and liability volume and duration. As such, this unit isolates and internalizes the interest rate risk stemming from changes in the slope and the curvature of the term structure, essentially representing a complex yield curve swap with time-varying notional principal, similarly to LBU and the ABU but, in contrast to them, based on market rates and not bank rates. For this, we require either perfect elasticity in the interbank market or that the bank’s funding requirements are small relative to the capacity of the interbank market, i.e. in any case the ALM can take net positive and negative position without affecting the market rate. In order to avoid strenuous positions that might expose the bank to liquidity risk, we further assume that the bank does not voluntarily grow the business volume on one side of its balance sheet wile leaving the other behind. Rather, asset and liability volume will balance in the long run with short-run deviations due to stochastically arriving transactions. This can be achieved by introducing a defined maximum difference ∆∗ for the difference between assets and liabilities, ∆∗ , ∆ = A − L ≤ ∆∗ , (13) which may be thought of as maximum imbalance allowed by the bank’s risk committee. Should it be reached, this condition will limit the further growth of the larger of assets or liabilities. Attaching a precise value to this limit will depend on the bank 25 at hand for valuation; for now, we confine ourselves to a general formalization, At+1 = Lt+1 = αA + βA At + γA XA + εA,t if ∆t+1 ≤ ∆∗ ∆∗ if ∆t+1 > ∆∗ αL + βL Lt + γL XL + εL,t if ∆t+1 ≤ ∆∗ ∆∗ if ∆t+1 > ∆∗ (14a) (14b) The ALM is slightly easier to value than either LBU or ABU. Like the other units, it can be modeled as a complex swap The ALM’s value can then be obtained by T −1 (rτ =2,t × At − rτ =1,t × Lt ) P∗ . (15) VALM = E0 g(t + 1) t=0 The ALM concentrates the beneficial effect of combining the activities of lending and borrowing, which have been argued to be manifold; for example, Diamond and Dybvig (1983) and Kashyap et al. (2002) propose that the combination offers a natural hedge against liquidity risk, see also Strahan (2005) for a recent survey of empirical evidence. In our model, this idea is intuitively exemplified by the swap. 5 Complete model in a continuous-time specification What remains is to tie all ends together and to derive a value for the entire bank consisting of the three business units. Although the discrete representation in the previous section is more intuitive for outlining the model, solving the valuation equations is easier in a continuous-time specification. 5.1 Valuation of the LBU For our exposition of the valuation procedure in continuous time, we can follow—up to a certain point—JvD’s general derivation of a closed-form solution for the valuation of non-maturing deposits. They employ very general autoregressive processes for the deposit volume growth and the deposit rate. For volume, they assume a process of the form L L L L d log Lt = α0 + α1 t + α2 rt dt + α3 drt , (16) and for the deposit rate, they set L L L drL,t = β0 + β1 rt dt + β2 drt , (17) where rt represents what we earlier have written as rτ =1,t , i.e. the index is dropped for ease of notation. Hence, both variables are depending on the market rate, rt and a trend growth rate which is represented by t. These processes are very general and do not include all the identified variables from our survey of deposit valuation 26 models. At this point, however, the focus is on the derivation of a closed-form solution and introducing additional factors would complicate this endeavaour. The solution to these SDEs are Lt = L0 exp and rL,t = r0 + L β0 t L α0 t + 2 Lt α1 t 2 + L α2 0 L rs ds + α3 (rt − r0 ) (18) t + L β1 0 L rs ds + β2 (rt − r0 ) (19) The extension of the discrete time valuation result for deposits in equation 8 is VL = P∗ E0 0 T Lt (rt − rL ) dt , g(t) (20) and, when substituting the solutions in 19 and 18 into the deposit valuation formula 20, we obtain P∗ 0 L L (rt − r0 − β0 t − β1 T L L L L0 exp α0 t + α1 t2 + α2 2 VL = E0 t r ds 0 s L + α3 (rt − r0 ) g(t) × t r ds 0 s L − β2 (rt − r0 ) g(t) dt (21) As already mentioned in the previous section, one can interpret this solution as a complex interest rate swap with stochastically changing principal, dependent on t both the the average, 0 rs ds, and level, rt , of past market rates. Next, we need to specify the dynamics of the interest rate. A common representation is a stochastic mean-reverting differential equation of the short rate r, which can be stated as ˜ drt = a (¯t − rt ) dt + σdW (t), r (22) and, applying the the term structure model of Heath et al. (1992), the solution to this SDE is given by rt = f (0, t) + σ 2 (e−at − 1) + 2a2 2 t ˜ σe−a(t−s) dW (s). 0 (23) Applying this to the deposit value in equation 21 and some tedious manipulations 27 yield the following solution:48 L L VL = L0 exp(−α3 r0 )(1 − β2 ) × L α1 t2 L L 2 L L ]M (t, α2 − 1, α3 )[µ(t) + σ2 (t)α3 + σ12 (t)(α2 − 1)]dt + 2 0 T αL t2 L L L L L L exp[α0 t + 1 ](−rL,0 − β0 t + β2 r0 )M (t, α2 − 1, α3 )dt +L0 exp[−α3 r0 ] 2 0 L L −L0 exp[−α3 r0 ]β1 × T αL t2 L L L 2 L L exp[α0 t + 1 ]M (t, α2 − 1, α3 )[µ2 (t) + σ1 (t)(α2 − 1) + σ12 (t)α3 ]dt, 2 0 where t t [σ 2 (1 − exp[−a(t − s)])2 ] 0 µ1 ≡ f (0, s)ds + , 2 0 t σ 2 (1 − exp[−a(t − s)])2 2 σ1 ≡ ds, 2 0 µ2 ≡ f (0, t) + σ12 , L exp[α0 t + t 2 σ2 ≡ 0 2 T (24) (25) (26) (27) (28) (29) σ 2 exp[−2a(t − s)]ds, σ12 σ (1 − exp[−at])2 ≡ , 2a2 2 2 2 2 σ1 (t)γ1 + 2σ12 (t)γ1 γ2 + σ2 (t)γ2 .(30) 2 It can be shown that this result allows a closed-form solution which has a rather bulky format, though. Jarrow and van Deventer (1998) provide a solution to these integrals in their appendix. Now, we have a closed-form solution for the deposit business. Whether one resorts to this solution or adds complexity with additional factors and increases empirical fit on the cost of having to rely on numerical methods is another crucial question which we will not discuss at this point, though. Although appealing, this solution is of little help for our problem of valuing the entire bank and we included it for exemplary reasons only. Implementing the model with a single-factor term structure model, as JvD do with a one-factor version of HJM, has the already mentioned drawbacks and does not offer yield curve dynamics rich enough to allow for the different relevant maturities within the bank. However, resorting to a two-factor model, which is necessary for the valuation of the entire bank, comes at the price of loosing the closed-form solution. Therefore, within the bank, we have to obtain the LBU’s value in numerical procedures. M (t, γ1 , γ2 ) ≡ exp µ1 (t)γ1 + µ2 (t)γ2 + 5.2 Valuation of the ABU The valuation of the ABU follows straightforwardly, mirroring the solution for the LBU’s value. Similarly, we can define general autoregressive processes for the evolu48 See Jarrow and van Deventer (1998), pp. 263–267 and appendix A.3 for a detailed derivation. 28 tion of the loan rate, rA and the asset volume, A, which suffices for the derivation of a expository solution but would await further refinement for an empirical implementation covering all relevant factors. Analogous to the previous processes, we define the asset volume to follow A A A A d log At = α0 + α1 t + α2 rt dt + α3 drt , (31) and the loan rate to behave according to A A A drA,t = β0 + β1 rt dt + β2 drt . (32) To properly value the ABU at the same time as the LBU, we have to rely on the dynamics of a two-factor model which restricts us from solving for an algebraic solution; rather, we have to rely in Monte Carlo simulations. 5.3 Valuation of the ALM T 0 The value of the ALM in continuous time is given by P VALM = E0 ∗ (rτ =2,t × At − rτ =1,t × Lt ) dt . g(t) (33) We were not able to derive a closed-form solution to this complex swap and have to rely on numerical procedures for its valuation. 5.4 Valuation of the entire bank For the valuation of such an interest-rate sensitive call option on the portfolio of the three business units, we have to rely on numerical methods. Given the fact that the drift and the volatilities of each new state variable do not contain each other, we have to resort to a trinomial lattice to match the required moments, and ensure recombination. Hull and White (1994) show how to construct such a tree in a trinomial framework. We are currently working on a trinomial tree representation. 6 Conclusion We argue that the problem of valuing a bank as a firm which is particularly exposed to interest rate risk has not been adequately solved in the literature so far. We propose a bank equity valuation model based on the contingent claims theory and derive the banking firm value as constituting of the value of three stylized business units, the asset business, liability business, and the asset-liability management. The value of each of these units can be derived in a risk-neutral valuation framework as equivalent to the costs of a hedging strategy that offsets the risk exposure but still allows for arbitrage profits. There exist several models for the valuation of banking products and we have exemplified our model using the deposit valuation model of 29 Jarrow and van Deventer (1998). Although our approach is workable with other models as well, the appeal of this particular deposit model is the availability of a closed-form solution for the economic value of bank assets and liabilities. Regarding our approach, some caveats and criticism are in order. The appeal of the just proposed valuation model is (a) that it includes interest rate risk but avoids interest rate forecasts, and (b) that it provides an elegant and comprehensive riskneutral framework which circumvents many of the problems of standard valuation approaches. A necessary assumption for this to work is the tradability of the hedging portfolio. We argue that such assets are continuously tradable, think e.g. of the markets for certificates of deposits or collateralized debt obligations. If one finds this assumption problematic, a possible alternative comes in the form of almost good deal boundaries. Cochrane and Sa´-Requejo (2001) propose these as a method a to value contingent claims for which a perfectly correlated asset is unavailable. The paper at hand merely sketches our approach and as such can be likened to the pizza Margarita of bank valuation: It’s got the basics ingredients but a little more toppings would be nice. We can think of several extensions to this basic model. For example, this framework should also be suitable to derive a default risk model of the bank, which we are currently attacking in a follow-on to this paper. Thinking in the same direction, a less crude balance sheet of the bank might enhance the insights of the model. One could allow for several different banking products on both sides of the balance sheet, also including reserve requirements. Besides, a more refined method for the measurement of credit risk of bank customers, i.e. the expected loss in the asset business, would certainly improve the model. Finally, a critical question is the inclusion of deposit insurance. It can be shown that the value of deposit insurance is akin to a put option and e.g. Ronn and Verma (1986) propose a valuation model for this. Although the benefits of this insurance accrue to the deposit holders and not to the shareholders of the bank, the latter might still be able to extract value from it, e.g. by assuming a more aggressive leverage on its balance sheet. Possible consequence of deposit insurance pose an interesting side issue. An empirical implementation of this model would be very interesting as well. For this, one would have to identify the relevant factors for the stochastic processes of the bank rate and business volume. So far, we are aware of only one study which applied a deposit valuation model to a single (anonymous) bank,49 suggesting that data availability might be an issue. One last interesting question is whether the bank’s spreads can be considered arbitrage profits. The answer to this will have far-reaching consequences for the valuation approach chosen. So far, we have assumed that these spreads do represent an arbitrage opportunity, although this valuation can only be exploited by banks. This clearly represents a simplification and one might ponder whether answering this question in the negative would make more sense. After all, this spread is no free lunch; at most, it can be considered an attractive lunch. In order to earn this spread, a bank has to incur the idiosyncratic business risk of the banking industry. 49 This is Janosi et al. (1999), who implement JvD’s model. 30 It has to expend on infrastructure and to sustain fixed cost of operations. Should competitive pressures erode the spread, the liability (or asset) volume, or both, the bank might not be able to support its fixed cost anymore. 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